Human languages share an optimal information transmission rate
Full statistical analysis and plots
Christophe Coupé, Dan Dediu and François Pellegrino
2019-03-02 15:57:44
Introduction
This Rmarkdown script (and corresponding TAB-separated CSV input data file InfoRateData.csv and the resulting HTML document) contain the full analysis and plotting code accompanying the paper Human languages share an optimal information transmission rate.
The data
For more information on the data, please see Oh (2015). There are in total 17 languages (see the Table below).
The Oral Corpus
The oral corpus is based on a subset of the Multext (Multilingual Text Tools and Corpora) parallel corpus (Campione & Véronis, 1998) in British English, German, and Italian. The material consists of 15 short texts of 3-5 semantically connected sentences carefully translated by a native speaker in each language.
For the other 14 languages, two of the authors supervised the translation and recording of new datasets. All participants were native speakers of the target language, with a focus on a specific variety of the language when possible – e.g. Mandarin spoken in Beijing, Serbian in Belgrade and Korean in Seoul. No strict control on age or on the speakers’ social diversity was performed, but speakers were mainly students or members of academic institutions. Speakers were asked to read three times (first silently and then loudly twice) each text. The texts were presented one by one on the screen in random order, in a self-paced reading paradigm. This way, speakers familiarized themselves with the text and reduce their reading errors. The second loud recording was analyzed in this study.
Text Corpus
Text datasets were acquired from various sources as illustrated in the Table below. After an initial data curation, each dataset was phonetically transcribed and automatically syllabified by a rule-based program written by one of the authors, except in the following cases:
- when syllabification was already provided with the dataset (English, French, German, and Vietnamese for the multisyllabic words);
- when the corpus was syllabified by an automatic grapheme-to-phoneme converter (Catalan, Spanish, and Thai).
Additionally, no syllabification was required for Sino-Tibetan languages (Cantonese and Mandarin Chinese) since one ideogram corresponds to one syllable.
| Language | Family | ISO 639-3 | Corpus |
|---|---|---|---|
| Basque | Basque | EUS | E-Hitz (Perea et al., 2006) |
| British English | Indo-European | ENG | WebCelex (MPI for Psycholinguistics) |
| Cantonese | Sino-Tibetan | YUE | A linguistic corpus of mid-20th century Hong Kong Cantonese |
| Catalan | Indo-European | CAT | Frequency dictionary (Zséder et al., 2012) |
| Finnish | Uralic | FIN | Finnish Parole Corpus |
| French | Indo-European | FRA | Lexique 3.80 (New et al., 2001) |
| German | Indo-European | DEU | WebCelex (MPI for Psycholinguistics) |
| Hungarian | Uralic | HUN | Hungarian National Corpus (Váradi, 2002) |
| Italian | Indo-European | ITA | The Corpus PAISÀ (Lyding et al., 2014) |
| Japanese | Japanese | JPN | Japanese Internet Corpus (Sharoff, 2006) |
| Korean | Korean | KOR | Leipzig Corpora Collection (LCC) |
| Mandarin Chinese | Sino-Tibetan | CMN | Chinese Internet Corpus (Sharoff, 2006) |
| Serbian | Indo-European | SRP | Frequency dictionary (Zséder et al., 2012) |
| Spanish | Indo-European | SPA | Frequency dictionary (Zséder et al., 2012) |
| Thai | Tai-Kadai | THA | Thai National Corpus (TNC) |
| Turkish | Turkic | TUR | Leipzig Corpora Collection (LCC) |
| Vietnamese | Austroasiatic | VIE | VNSpeechCorpus (Le et al., 2004) |
Dataset structure
The data is structured as follows:
- Language is the unique language ISO 639-3 ID with 17 possible values “CAT”, “CMN”, “DEU”, “ENG”, “EUS”, “FIN”, “FRA”, “HUN”, “ITA”, “JPN”, “KOR”, “SPA”, “SRP”, “THA”, “TUR”, “VIE”, “YUE”;
- Language family is the language family with 9 possible values “Austroasiatic”, “Basque”, “Indo-European”, “Japanese”, “Korean”, “Sino-Tibetan”, “Tai-Kadai”, “Turkic”, “Uralic”;
- Text is the text identifier (similar across languages and speakers) with 15 possible values “O1”, “O2”, “O3”, “O4”, “O6”, “O8”, “O9”, “P0”, “P1”, “P2”, “P3”, “P8”, “P9”, “Q0”, “Q1”;
- Speaker is the unique speaker’s ID (there are 170 in the dataset);
- Sex is the speaker’s sex;
- Age is the speaker’s age (available only for a subset of 132 speakers);
- Duration is the texts’ duration in seconds as spoken by the given speaker (pauses longer than 150ms being excluded);
- NS is the texts’ number of of syllables (the same for a given text in a given language);
- SR is the speech rate (syllables/second) for a given text spoken by a given speaker;
- ShE (resp. ID) is the first-order (resp. second-order) language-level entropy estimate;
- ShIR and IR are text * speaker properties obtained by multiplying SR by ShE and ID respectively.
Exploratory plots and summaries
Speaker characteristics
| Lng | # spkrs | % fem | # age | mean(age) | sd(age) | actual ages |
|---|---|---|---|---|---|---|
| CAT | 10 | 50 | 10 | 35.4 | 9.2 | (21, 28, 28, 29, 31, 39, 42, 42, 44, 50) |
| CMN | 10 | 50 | 9 | 23.1 | 4.5 | (19, 19, 19, 19, 24, 24, 25, 28, 31) |
| DEU | 10 | 50 | 0 | NaN | NaN | () |
| ENG | 10 | 50 | 0 | NaN | NaN | () |
| EUS | 10 | 50 | 10 | 28.0 | 4.9 | (19, 22, 26, 27, 28, 29, 30, 31, 32, 36) |
| FIN | 10 | 50 | 10 | 33.2 | 11.0 | (16, 22, 26, 28, 30, 35, 37, 41, 45, 52) |
| FRA | 10 | 50 | 10 | 32.5 | 7.7 | (24, 25, 25, 27, 28, 36, 36, 37, 41, 46) |
| HUN | 10 | 50 | 10 | 39.3 | 15.8 | (17, 27, 27, 31, 33, 39, 42, 51, 57, 69) |
| ITA | 10 | 50 | 0 | NaN | NaN | () |
| JPN | 10 | 50 | 10 | 30.6 | 12.8 | (20, 20, 21, 22, 22, 28, 29, 40, 51, 53) |
| KOR | 10 | 50 | 10 | 28.6 | 10.6 | (16, 19, 19, 19, 28, 31, 33, 35, 36, 50) |
| SPA | 10 | 50 | 10 | 33.7 | 10.1 | (21, 22, 26, 28, 30, 32, 42, 42, 44, 50) |
| SRP | 10 | 50 | 10 | 30.6 | 7.8 | (19, 21, 23, 30, 31, 32, 34, 34, 38, 44) |
| THA | 10 | 50 | 10 | 30.1 | 5.7 | (23, 23, 27, 28, 30, 31, 31, 32, 33, 43) |
| TUR | 10 | 50 | 7 | 32.6 | 7.2 | (24, 25, 30, 31, 37, 37, 44) |
| VIE | 10 | 50 | 6 | 27.2 | 4.1 | (21, 25, 26, 28, 31, 32) |
| YUE | 10 | 50 | 10 | 22.0 | 1.5 | (20, 20, 21, 21, 22, 22, 23, 23, 24, 24) |
NS
NS: exploratory plots.
mean=101.217, median=100, sd=24.694, CV=0.244, min=49, max=162, kurtosis=2.485, skewness=0.178.SR
SR: exploratory plots.
SR per speaker.
SR by Sex and Age across Languages.
SR by Sex, Age and Language.
SR by language.
mean=6.612, median=6.76, sd=1.133, CV=0.171, min=3.59, max=9.38, kurtosis=2.382, skewness=-0.197.ShE and ID
ShE and ID: exploratory plots.
ShE vs ID.
Pearson's product-moment correlation
data: tmp1$ShE and tmp1$ID
t = 2.0326, df = 15, p-value = 0.06019
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
-0.02052914 0.77274779
sample estimates:
cor
0.4647009
Spearman's rank correlation rho
data: tmp1$ShE and tmp1$ID
S = 451.88, p-value = 0.07259
alternative hypothesis: true rho is not equal to 0
sample estimates:
rho
0.4462208
Paired t-test
data: tmp1$ShE and tmp1$ID
t = 11.635, df = 16, p-value = 3.213e-09
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
2.158040 3.119607
sample estimates:
mean of the differences
2.638824 ShE:
mean=8.619, median=8.69, sd=0.906, CV=0.105, min=6.07, max=9.83, kurtosis=4.662, skewness=-1.124.ID:
mean=6.009, median=5.56, sd=0.883, CV=0.147, min=4.83, max=8.02, kurtosis=2.521, skewness=0.741.ShIR and IR
ShIR: exploratory plots.
ShIR per speaker.
ShIR by Sex and Age across Languages.
ShIR by Sex, Age and Language.
ShIR by language.
IR: exploratory plots.
IR per speaker.
IR by Sex and Age across Languages.
IR by Sex, Age and Language.
IR by language.
ShIR:
mean=56.525, median=56.92, sd=9.217, CV=0.163, min=32.77, max=83.04, kurtosis=2.335, skewness=0.048.IR:
mean=39.033, median=39.05, sd=4.945, CV=0.127, min=25.64, max=56.3, kurtosis=3.422, skewness=0.241.SR and ID
SR vs ID
Pearson's product-moment correlation
data: info.rate.data$SR and info.rate.data$ID
t = -46.635, df = 2263, p-value < 2.2e-16
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
-0.7204624 -0.6784227
sample estimates:
cor
-0.7000486
Spearman's rank correlation rho
data: info.rate.data$SR and info.rate.data$ID
S = 3315700000, p-value < 2.2e-16
alternative hypothesis: true rho is not equal to 0
sample estimates:
rho
-0.7120735 Linear Mixed Modells (LMMs)
Linear ICCs
| Level-1 factor (f) | ICC |
|---|---|
| Text | 0.30 |
| Language | 0.54 |
| Speaker | 0.00 |
| Level-1 factor (f) | ICC |
|---|---|
| Text | 0.01 |
| Language | 0.62 |
| Speaker | 0.29 |
| Level-1 factor (f) | ICC |
|---|---|
| Text | 0.01 |
| Language | 0.56 |
| Speaker | 0.34 |
| Level-1 factor (f) | ICC |
|---|---|
| Text | 0.02 |
| Language | 0.36 |
| Speaker | 0.49 |
SR
Comparing models
| model | AIC | BIC |
|---|---|---|
| 1 + (1 | Text) + (1 | Language) + (1 | Speaker) | 1998.03 | 2026.65 |
| 1 + (1 | Language) + (1 | Speaker) | 2269.3 | 2292.2 |
| 1 + (1 | Text) + (1 | Speaker) | 2131.96 | 2154.86 |
| 1 + (1 | Text) + (1 | Language) | 4490.99 | 4513.89 |
| 1 + Sex + (1 | Text) + (1 | Language) + (1 | Speaker) | 1990.57 | 2024.92 |
| 1 + Sex + (1 | Language) + (1 | Speaker) | 2261.69 | 2290.31 |
| 1 + Sex + (1 | Text) + (1 | Speaker) | 2131.72 | 2160.35 |
| 1 + Sex + (1 | Text) + (1 | Language) | 4364.38 | 4393.01 |
Residuals and random effects
We consider here the full model SR ~ 1 + Sex + (1|Text) + (1|Language) + (1|Speaker).
IR
Comparing models
| model | AIC | BIC |
|---|---|---|
| 1 + (1 | Text) + (1 | Language) + (1 | Speaker) | 10041.03 | 10069.65 |
| 1 + (1 | Language) + (1 | Speaker) | 10307.63 | 10330.53 |
| 1 + (1 | Text) + (1 | Speaker) | 10093.09 | 10115.99 |
| 1 + (1 | Text) + (1 | Language) | 12574.43 | 12597.33 |
| 1 + Sex + (1 | Text) + (1 | Language) + (1 | Speaker) | 10029.53 | 10063.88 |
| 1 + Sex + (1 | Language) + (1 | Speaker) | 10295.98 | 10324.6 |
| 1 + Sex + (1 | Text) + (1 | Speaker) | 10086.35 | 10114.97 |
| 1 + Sex + (1 | Text) + (1 | Language) | 12440.04 | 12468.67 |
Residuals and random effects
We consider here the full model IR ~ 1 + Sex + (1|Text) + (1|Language) + (1|Speaker).
Generalized Additive Models for Location, Scale and Shape (GAMLSS)
We will use a Gaussian distribution (with fixed or modelled variance).
SR
Fixed or modelled σ (variance)
Fixed σ
******************************************************************
Summary of the Quantile Residuals
mean = 6.666774e-05
variance = 1.000442
coef. of skewness = -0.0552563
coef. of kurtosis = 3.323173
Filliben correlation coefficient = 0.9990793
******************************************************************
Deviance= 1048.516
AIC= 1447.603 Modelled σ
******************************************************************
Summary of the Quantile Residuals
mean = 0.001738762
variance = 1.000438
coef. of skewness = -0.02112376
coef. of kurtosis = 2.757837
Filliben correlation coefficient = 0.9991447
******************************************************************
Deviance= 719.0835
AIC= 1299.257 The distribution of the residuals is less heteroscedastic than before and the fit to the data better. The full summary of the model is:
******************************************************************
Family: c("NO", "Normal")
Call: gamlss(formula = SR ~ 1 + Sex + random(Text) + random(Language) + random(Speaker), sigma.formula = ~1 + Sex + random(Text) + random(Language) + random(Speaker), family = NO(mu.link = "identity"),
data = d, control = gamlss.control(n.cyc = 800, trace = FALSE), i.control = glim.control(bf.cyc = 800))
Fitting method: RS()
------------------------------------------------------------------
Mu link function: identity
Mu Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 6.453290 0.007618 847.14 <2e-16 ***
SexM 0.320159 0.011487 27.87 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
------------------------------------------------------------------
Sigma link function: log
Sigma Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.30948 0.02098 -62.417 < 2e-16 ***
SexM 0.11194 0.02972 3.767 0.00017 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
------------------------------------------------------------------
NOTE: Additive smoothing terms exist in the formulas:
i) Std. Error for smoothers are for the linear effect only.
ii) Std. Error for the linear terms maybe are not accurate.
------------------------------------------------------------------
No. of observations in the fit: 2265
Degrees of Freedom for the fit: 290.0869
Residual Deg. of Freedom: 1974.913
at cycle: 48
Global Deviance: 719.0835
AIC: 1299.257
SBC: 2960.101
******************************************************************Random effects
μ
Text
Random effects fit using the gamlss function random()
Degrees of Freedom for the fit : 14.3977
Random effect parameter sigma_b: 0.108314
Smoothing parameter lambda : 85.7824 Language
Random effects fit using the gamlss function random()
Degrees of Freedom for the fit : 16.9836
Random effect parameter sigma_b: 0.87066
Smoothing parameter lambda : 1.32915 Speaker
Random effects fit using the gamlss function random()
Degrees of Freedom for the fit : 165.2778
Random effect parameter sigma_b: 0.555308
Smoothing parameter lambda : 3.49815 σ
Text
Random effects fit using the gamlss function random()
Degrees of Freedom for the fit : 0.583681
Random effect parameter sigma_b: 0.0108169
Smoothing parameter lambda : 7523.57 Language
Random effects fit using the gamlss function random()
Degrees of Freedom for the fit : 14.91065
Random effect parameter sigma_b: 0.161228
Smoothing parameter lambda : 34.0803 Speaker
Random effects fit using the gamlss function random()
Degrees of Freedom for the fit : 73.93344
Random effect parameter sigma_b: 0.165342
Smoothing parameter lambda : 33.2784 IR
Fixed or modelled σ (variance)
Fixed σ
******************************************************************
Summary of the Quantile Residuals
mean = 0.0002550467
variance = 1.000442
coef. of skewness = 0.01302313
coef. of kurtosis = 3.360878
Filliben correlation coefficient = 0.9990567
******************************************************************
Deviance= 9108.279
AIC= 9507.442 Modelled σ
******************************************************************
Summary of the Quantile Residuals
mean = 0.001906696
variance = 1.000449
coef. of skewness = -0.02187398
coef. of kurtosis = 2.755234
Filliben correlation coefficient = 0.9991259
******************************************************************
Deviance= 8782.878
AIC= 9364.641 Again, this is a better fit to the data. The full summary of the model is:
******************************************************************
Family: c("NO", "Normal")
Call: gamlss(formula = IR ~ 1 + Sex + random(Text) + random(Language) + random(Speaker), sigma.formula = ~1 + Sex + random(Text) + random(Language) + random(Speaker), family = NO(mu.link = "identity"),
data = d, control = gamlss.control(n.cyc = 800, trace = FALSE), i.control = glim.control(bf.cyc = 800))
Fitting method: RS()
------------------------------------------------------------------
Mu link function: identity
Mu Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 38.08916 0.04521 842.56 <2e-16 ***
SexM 1.89529 0.06851 27.66 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
------------------------------------------------------------------
Sigma link function: log
Sigma Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.47207 0.02098 22.501 < 2e-16 ***
SexM 0.11385 0.02972 3.831 0.000131 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
------------------------------------------------------------------
NOTE: Additive smoothing terms exist in the formulas:
i) Std. Error for smoothers are for the linear effect only.
ii) Std. Error for the linear terms maybe are not accurate.
------------------------------------------------------------------
No. of observations in the fit: 2265
Degrees of Freedom for the fit: 290.8814
Residual Deg. of Freedom: 1974.119
at cycle: 7
Global Deviance: 8782.878
AIC: 9364.641
SBC: 11030.03
******************************************************************Random effects
μ
Text
Random effects fit using the gamlss function random()
Degrees of Freedom for the fit : 14.42046
Random effect parameter sigma_b: 0.658373
Smoothing parameter lambda : 2.32185 Language
Random effects fit using the gamlss function random()
Degrees of Freedom for the fit : 16.9518
Random effect parameter sigma_b: 3.06988
Smoothing parameter lambda : 0.106913 Speaker
Random effects fit using the gamlss function random()
Degrees of Freedom for the fit : 165.1517
Random effect parameter sigma_b: 3.31521
Smoothing parameter lambda : 0.0981439 σ
Text
Random effects fit using the gamlss function random()
Degrees of Freedom for the fit : 0.09852989
Random effect parameter sigma_b: 0.00428681
Smoothing parameter lambda : 47821.8 Language
Random effects fit using the gamlss function random()
Degrees of Freedom for the fit : 14.96383
Random effect parameter sigma_b: 0.163535
Smoothing parameter lambda : 33.0774 Speaker
Random effects fit using the gamlss function random()
Degrees of Freedom for the fit : 75.29513
Random effect parameter sigma_b: 0.168066
Smoothing parameter lambda : 32.181 Modelling the relationship between SR and ID
Let’s model SR with ID as an additional predictor (fixed effect). N.B. In this case, we must drop Language as a random effect, since each language has, by definition, only one value of ID.
******************************************************************
Family: c("NO", "Normal")
Call: gamlss(formula = SR ~ 1 + ID + Sex + random(Text) + random(Speaker), sigma.formula = ~1 + ID + Sex + random(Text) + random(Speaker), family = NO(mu.link = "identity"), data = d, control = gamlss.control(n.cyc = 800,
trace = FALSE), i.control = glim.control(bf.cyc = 800))
Fitting method: RS()
------------------------------------------------------------------
Mu link function: identity
Mu Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 11.794174 0.036777 320.69 <2e-16 ***
ID -0.888238 0.005842 -152.06 <2e-16 ***
SexM 0.310995 0.011382 27.32 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
------------------------------------------------------------------
Sigma link function: log
Sigma Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.79946 0.10475 -7.632 3.57e-14 ***
ID -0.08642 0.01705 -5.070 4.35e-07 ***
SexM 0.10951 0.02972 3.685 0.000235 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
------------------------------------------------------------------
NOTE: Additive smoothing terms exist in the formulas:
i) Std. Error for smoothers are for the linear effect only.
ii) Std. Error for the linear terms maybe are not accurate.
------------------------------------------------------------------
No. of observations in the fit: 2265
Degrees of Freedom for the fit: 293.322
Residual Deg. of Freedom: 1971.678
at cycle: 18
Global Deviance: 686.5481
AIC: 1273.192
SBC: 2952.558
******************************************************************
******************************************************************
Summary of the Quantile Residuals
mean = 0.001881969
variance = 1.000438
coef. of skewness = -0.02304788
coef. of kurtosis = 2.677419
Filliben correlation coefficient = 0.9989511
******************************************************************
Deviance= 686.5481
AIC= 1273.192 Adding ID as a predictor improves the fits (as judged by AIC). There is a negative estimate for ID, but significance is difficult to assess with GAMLSS model involving smoothing functions. However, also using a simple lmer model we have a significant effect of ID:
Type III Analysis of Variance Table with Satterthwaite's method
Sum Sq Mean Sq NumDF DenDF F value Pr(>F)
ID 16.4136 16.4136 1 166.55 162.5728 < 2.2e-16 ***
Sex 0.8047 0.8047 1 166.63 7.9707 0.005334 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Random effects
μ
Text
Random effects fit using the gamlss function random()
Degrees of Freedom for the fit : 14.42354
Random effect parameter sigma_b: 0.10986
Smoothing parameter lambda : 83.3869 Speaker
Random effects fit using the gamlss function random()
Degrees of Freedom for the fit : 167.487
Random effect parameter sigma_b: 0.74601
Smoothing parameter lambda : 1.94032 σ
Text
Random effects fit using the gamlss function random()
Degrees of Freedom for the fit : 1.916828
Random effect parameter sigma_b: 0.0201916
Smoothing parameter lambda : 2061.61 Speaker
Random effects fit using the gamlss function random()
Degrees of Freedom for the fit : 103.4947
Random effect parameter sigma_b: 0.234219
Smoothing parameter lambda : 16.0415 Do Age and Sex matter?
Linear mixed model fit by REML. t-tests use Satterthwaite's method ['lmerModLmerTest']
Formula: SR ~ Age * Sex + (1 | Text) + (1 | Language)
Data: info.rate.data
REML criterion at convergence: 3541.9
Scaled residuals:
Min 1Q Median 3Q Max
-2.8993 -0.6397 0.0095 0.6216 3.2123
Random effects:
Groups Name Variance Std.Dev.
Text (Intercept) 0.01201 0.1096
Language (Intercept) 1.01269 1.0063
Residual 0.33375 0.5777
Number of obs: 1959, groups: Text, 15; Language, 14
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 6.663e+00 2.791e-01 1.499e+01 23.876 2.43e-13 ***
Age -6.219e-03 2.177e-03 1.930e+03 -2.857 0.00432 **
SexM 1.808e-01 8.874e-02 1.928e+03 2.037 0.04174 *
Age:SexM 3.058e-03 2.770e-03 1.928e+03 1.104 0.26986
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr) Age SexM
Age -0.237
SexM -0.176 0.682
Age:SexM 0.170 -0.716 -0.956Linear mixed model fit by REML. t-tests use Satterthwaite's method ['lmerModLmerTest']
Formula: IR ~ Age * Sex + (1 | Text) + (1 | Language)
Data: info.rate.data
REML criterion at convergence: 10507.9
Scaled residuals:
Min 1Q Median 3Q Max
-2.8289 -0.6365 0.0016 0.6136 3.5323
Random effects:
Groups Name Variance Std.Dev.
Text (Intercept) 0.3901 0.6246
Language (Intercept) 10.1922 3.1925
Residual 11.8777 3.4464
Number of obs: 1959, groups: Text, 15; Language, 14
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 39.33279 0.96043 20.45736 40.953 < 2e-16 ***
Age -0.04214 0.01298 1933.79776 -3.247 0.00119 **
SexM 1.08599 0.52934 1929.48938 2.052 0.04034 *
Age:SexM 0.01915 0.01653 1929.60714 1.159 0.24671
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr) Age SexM
Age -0.411
SexM -0.305 0.682
Age:SexM 0.294 -0.716 -0.956So, it seems * Age* and Sex are both worth including in our models (even if we have to discard quite a bit of data because of missing Age info). (In fact, the effect of Age seems more significant than that of Sex.)
In the following, we investigate if Age does matter when using GAMLSS modelling…
GAMLSS with Age
Bescause there is missing data fro Age, and because the GAMLSS models require no missing data, we will fit the models with Age (and its interaction with Sex) on the subset of the data that contains only those speakers with Age info. To make comparability possible, we also fit the same models but without Age on the exact same subset of the data.
SR
Fixed or modelled σ (variance)
Fixed σ
******************************************************************
Summary of the Quantile Residuals
mean = -6.761517e-05
variance = 1.000511
coef. of skewness = -0.07716915
coef. of kurtosis = 3.390812
Filliben correlation coefficient = 0.998745
******************************************************************
The model including Age * Sex is:
******************************************************************
Family: c("NO", "Normal")
Call: gamlss(formula = SR ~ 1 + Sex * Age + random(Text) + random(Language) + random(Speaker), family = NO(mu.link = "identity"), data = info.rate.data.for.age, control = gamlss.control(n.cyc = 800,
trace = FALSE), i.control = glim.control(bf.cyc = 800))
Fitting method: RS()
------------------------------------------------------------------
Mu link function: identity
Mu Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 6.611763 0.033552 197.060 < 2e-16 ***
SexM 0.165405 0.045837 3.609 0.000316 ***
Age -0.003179 0.001050 -3.029 0.002487 **
SexM:Age 0.003528 0.001428 2.470 0.013601 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
------------------------------------------------------------------
Sigma link function: log
Sigma Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.18825 0.01598 -74.38 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
------------------------------------------------------------------
NOTE: Additive smoothing terms exist in the formulas:
i) Std. Error for smoothers are for the linear effect only.
ii) Std. Error for the linear terms maybe are not accurate.
------------------------------------------------------------------
No. of observations in the fit: 1959
Degrees of Freedom for the fit: 161.6144
Residual Deg. of Freedom: 1797.386
at cycle: 33
Global Deviance: 903.8325
AIC: 1227.061
SBC: 2128.901
******************************************************************The compared models are:
| Model | Deviance | AIC |
|---|---|---|
| Age * Sex | 903.8 | 1227.1 |
| Age + Sex | 903.8 | 1225.1 |
| Sex | 903.8 | 1223.1 |
So, even if Age has a significant (negative) effect and interaction with Sex (positive for males), adding it does not seem to be warranted here…
Modelled σ
******************************************************************
Summary of the Quantile Residuals
mean = 0.0022763
variance = 1.000505
coef. of skewness = -0.03406773
coef. of kurtosis = 2.772961
Filliben correlation coefficient = 0.9990489
******************************************************************
The model including Age * Sex is:
******************************************************************
Family: c("NO", "Normal")
Call: gamlss(formula = SR ~ 1 + Sex * Age + random(Text) + random(Language) + random(Speaker), sigma.formula = ~1 + Sex * Age + random(Text) + random(Language) + random(Speaker),
family = NO(mu.link = "identity"), data = info.rate.data.for.age, control = gamlss.control(n.cyc = 800, trace = FALSE), i.control = glim.control(bf.cyc = 800))
Fitting method: RS()
------------------------------------------------------------------
Mu link function: identity
Mu Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 6.6375862 0.0269709 246.101 < 2e-16 ***
SexM 0.2126714 0.0400028 5.316 1.20e-07 ***
Age -0.0040916 0.0008478 -4.826 1.52e-06 ***
SexM:Age 0.0019632 0.0012742 1.541 0.124
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
------------------------------------------------------------------
Sigma link function: log
Sigma Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.381266 0.080544 -17.149 <2e-16 ***
SexM 0.039702 0.108394 0.366 0.714
Age 0.002071 0.002525 0.820 0.412
SexM:Age 0.002463 0.003381 0.728 0.466
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
------------------------------------------------------------------
NOTE: Additive smoothing terms exist in the formulas:
i) Std. Error for smoothers are for the linear effect only.
ii) Std. Error for the linear terms maybe are not accurate.
------------------------------------------------------------------
No. of observations in the fit: 1959
Degrees of Freedom for the fit: 242.0172
Residual Deg. of Freedom: 1716.983
at cycle: 93
Global Deviance: 626.8351
AIC: 1110.87
SBC: 2461.371
******************************************************************The compared models are:
| Model | Deviance | AIC |
|---|---|---|
| Age * Sex | 626.8 | 1110.9 |
| Age + Sex | 627.3 | 1107.1 |
| Sex | 626.8 | 1104 |
So, even if Age has a significant (negative) effect (but no interaction with Sex), adding it does not seem to be warranted here either…
The distribution of the residuals is less heteroscedastic than before and the fit to the data better.
Summary
Thus, for SR, even if there is a hint that Age might affect it negatively (and there might also be an interaction with Sex with a positive effect for males), overall, the various fit indices do not warrant its inclusion in the GAMLSS models.
IR
Fixed or modelled σ (variance)
Fixed σ
******************************************************************
Summary of the Quantile Residuals
mean = 0.000175519
variance = 1.000511
coef. of skewness = -0.01914536
coef. of kurtosis = 3.370931
Filliben correlation coefficient = 0.9989951
******************************************************************
The model including Age * Sex is:
******************************************************************
Family: c("NO", "Normal")
Call: gamlss(formula = IR ~ 1 + Sex * Age + random(Text) + random(Language) + random(Speaker), family = NO(mu.link = "identity"), data = info.rate.data.for.age, control = gamlss.control(n.cyc = 800,
trace = FALSE), i.control = glim.control(bf.cyc = 800))
Fitting method: RS()
------------------------------------------------------------------
Mu link function: identity
Mu Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 39.626506 0.196112 202.061 < 2e-16 ***
SexM 0.082709 0.267917 0.309 0.758
Age -0.053300 0.006135 -8.688 < 2e-16 ***
SexM:Age 0.050696 0.008348 6.073 1.53e-09 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
------------------------------------------------------------------
Sigma link function: log
Sigma Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.57734 0.01598 36.14 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
------------------------------------------------------------------
NOTE: Additive smoothing terms exist in the formulas:
i) Std. Error for smoothers are for the linear effect only.
ii) Std. Error for the linear terms maybe are not accurate.
------------------------------------------------------------------
No. of observations in the fit: 1959
Degrees of Freedom for the fit: 161.7831
Residual Deg. of Freedom: 1797.217
at cycle: 3
Global Deviance: 7821.4
AIC: 8144.966
SBC: 9047.746
******************************************************************The compared models are:
| Model | Deviance | AIC |
|---|---|---|
| Age * Sex | 7821.4 | 8145 |
| Age + Sex | 7821.4 | 8143 |
| Sex | 7821.4 | 8141 |
So, even if Age has a significant (negative) effect and interaction with Sex (positive for males) – interestingly, in this case the main effect of Sex disappears –, adding it does not seem to be warranted…
Modelled σ
******************************************************************
Summary of the Quantile Residuals
mean = 0.001952693
variance = 1.000561
coef. of skewness = -0.03448375
coef. of kurtosis = 2.780376
Filliben correlation coefficient = 0.9990344
******************************************************************
The model including Age * Sex is:
******************************************************************
Family: c("NO", "Normal")
Call: gamlss(formula = IR ~ 1 + Sex * Age + random(Text) + random(Language) + random(Speaker), sigma.formula = ~1 + Sex * Age + random(Text) + random(Language) + random(Speaker),
family = NO(mu.link = "identity"), data = info.rate.data.for.age, control = gamlss.control(n.cyc = 800, trace = FALSE), i.control = glim.control(bf.cyc = 800))
Fitting method: RS()
------------------------------------------------------------------
Mu link function: identity
Mu Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 39.593055 0.157762 250.966 < 2e-16 ***
SexM 0.179628 0.233445 0.769 0.442
Age -0.052410 0.004872 -10.757 < 2e-16 ***
SexM:Age 0.047581 0.007310 6.509 9.9e-11 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
------------------------------------------------------------------
Sigma link function: log
Sigma Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.4322689 0.0804947 5.370 8.94e-08 ***
SexM 0.0240788 0.1084567 0.222 0.824
Age 0.0007877 0.0025235 0.312 0.755
SexM:Age 0.0029496 0.0033829 0.872 0.383
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
------------------------------------------------------------------
NOTE: Additive smoothing terms exist in the formulas:
i) Std. Error for smoothers are for the linear effect only.
ii) Std. Error for the linear terms maybe are not accurate.
------------------------------------------------------------------
No. of observations in the fit: 1959
Degrees of Freedom for the fit: 242.0359
Residual Deg. of Freedom: 1716.964
at cycle: 5
Global Deviance: 7559.015
AIC: 8043.087
SBC: 9393.693
******************************************************************The compared models are:
| Model | Deviance | AIC |
|---|---|---|
| Age * Sex | 7559 | 8043.1 |
| Age + Sex | 7559.5 | 8039.4 |
| Sex | 7558.6 | 8036.1 |
So, even if Age has a significant (negative) effect and interaction with Sex (positive for males) – interestingly, in this case the main effect of Sex disappears –, adding it does not seem to be warranted…
The distribution of the residuals is less heteroscedastic than before and the fit to the data better.
Summary
Thus, while for IR the hint that Age has a negative main effect and interacts with Sex (with a positive effect for males, containing the whole effect of Sex) is much stronger, the various fit indices do not warrant its inclusion in the GAMLSS models.
Modelling the relationship between SR and ID
******************************************************************
Summary of the Quantile Residuals
mean = 0.001627695
variance = 1.000508
coef. of skewness = -0.04476961
coef. of kurtosis = 2.716779
Filliben correlation coefficient = 0.99893
******************************************************************
The model including Age * Sex is:
******************************************************************
Family: c("NO", "Normal")
Call: gamlss(formula = SR ~ 1 + ID + Sex * Age + random(Text) + random(Speaker), sigma.formula = ~1 + ID + Sex * Age + random(Text) + random(Speaker), family = NO(mu.link = "identity"),
data = info.rate.data.for.age, control = gamlss.control(n.cyc = 800, trace = FALSE), i.control = glim.control(bf.cyc = 800))
Fitting method: RS()
------------------------------------------------------------------
Mu link function: identity
Mu Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 12.6730928 0.0541143 234.191 <2e-16 ***
ID -0.9841747 0.0069997 -140.602 <2e-16 ***
SexM -0.0687297 0.0401500 -1.712 0.0871 .
Age -0.0101486 0.0008593 -11.811 <2e-16 ***
SexM:Age 0.0109265 0.0012842 8.509 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
------------------------------------------------------------------
Sigma link function: log
Sigma Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.7738535 0.1464447 -5.284 1.42e-07 ***
ID -0.0931526 0.0192597 -4.837 1.44e-06 ***
SexM -0.0109575 0.1092433 -0.100 0.920
Age 0.0001475 0.0025388 0.058 0.954
SexM:Age 0.0041300 0.0034099 1.211 0.226
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
------------------------------------------------------------------
NOTE: Additive smoothing terms exist in the formulas:
i) Std. Error for smoothers are for the linear effect only.
ii) Std. Error for the linear terms maybe are not accurate.
------------------------------------------------------------------
No. of observations in the fit: 1959
Degrees of Freedom for the fit: 237.3435
Residual Deg. of Freedom: 1721.656
at cycle: 9
Global Deviance: 606.1764
AIC: 1080.863
SBC: 2405.285
******************************************************************The compared models are:
| Model | Deviance | AIC |
|---|---|---|
| Age * Sex | 606.2 | 1080.9 |
| Age + Sex | 606.4 | 1077.2 |
| Sex | 606.4 | 1073.5 |
Clearly, adding Age is not warranted here…
As above, we also looked a the simple lmer model:
The compared models are:
| Model | AIC |
|---|---|
| Age * Sex | 1700.1 |
| Age + Sex | 1691.9 |
| Sex | 1682.2 |
Interestingly, here the best model (as suggested by AIC) is the one with both Age and Sex (but no interaaction), but, as can be seen below, actually the effect of Age is not significant (p = 0.482), suggesting that, in fact, the best model is still the one not including Age:
Linear mixed model fit by REML. t-tests use Satterthwaite's method ['lmerModLmerTest']
Formula: SR ~ 1 + ID + Sex + Age + (1 | Text) + (1 | Speaker)
Data: info.rate.data.for.age
REML criterion at convergence: 1677.9
Scaled residuals:
Min 1Q Median 3Q Max
-3.9435 -0.6442 0.0340 0.5995 3.0185
Random effects:
Groups Name Variance Std.Dev.
Speaker (Intercept) 0.50570 0.7111
Text (Intercept) 0.01462 0.1209
Residual 0.10011 0.3164
Number of obs: 1959, groups: Speaker, 132; Text, 15
Fixed effects:
Estimate Std. Error df t value Pr(>|t|)
(Intercept) 12.458038 0.526996 128.797113 23.640 <2e-16 ***
ID -0.976300 0.075151 127.920530 -12.991 <2e-16 ***
SexM 0.271787 0.124633 127.912972 2.181 0.031 *
Age -0.004619 0.006545 127.895657 -0.706 0.482
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Correlation of Fixed Effects:
(Intr) ID SexM
ID -0.910
SexM -0.114 -0.005
Age -0.521 0.167 -0.005Conclusions about Age
Within the limits of our reduced dataset (containing only 132 speakers with Age info), we found the following:
When modelling SR and IR with GAMLSS, while there are hints that Age has, for both, overall:
- a negative main effect (i.e., older speakers of both sexes are slower and transmit less information per unit time across texts and languages), and
- an interaction with Sex (positive for males, i.e., males are less negatively affected by age that females across texts and languages),
it does not seem warranted to include it in these models.
When modelling the relationship between SR and ID, this negative relationship:
- seems strengthened with increasing Age (i.e., older speakers of both sexes shows a stronger negative relationship between SR and ID), and
- there is an interaction with Sex (for females, this effect of Age is stronger than for males),
but, alas, the inclusion of Age is not warranted in the GAMLSS model, nor (really) in the simpler LMER model.
Thus, while Age seems to negatively influence (in a sex-dependent manner) both SR and IR, as well as strengthen the negative relationship between them, its effects are far from clear in the current dataset.
Mixture of Gaussians
In what follows, mixing probabilities are independent from factors such as Sex.
SR
Between 1 and 5 Gaussian distributions:
1 component
Mixing Family: "NO"
Fitting method: EM algorithm
Call: gamlssMX(formula = SR ~ 1, family = NO, K = 1, data = d, plot = FALSE)
Mu Coefficients for model: 1
(Intercept)
6.612
Sigma Coefficients for model: 1
(Intercept)
0.1248
Estimated probabilities: 1
Degrees of Freedom for the fit: 2 Residual Deg. of Freedom 2263
Global Deviance: 6993.16
AIC: 6997.16
SBC: 7008.61 2 components
Mixing Family: c("NO", "NO")
Fitting method: EM algorithm
Call: gamlssMX(formula = SR ~ 1, family = NO, K = 2, data = d, plot = FALSE)
Mu Coefficients for model: 1
(Intercept)
5.293
Sigma Coefficients for model: 1
(Intercept)
-0.4553
Mu Coefficients for model: 2
(Intercept)
7.157
Sigma Coefficients for model: 2
(Intercept)
-0.2298
Estimated probabilities: 0.2925825 0.7074175
Degrees of Freedom for the fit: 5 Residual Deg. of Freedom 2260
Global Deviance: 6861.89
AIC: 6871.89
SBC: 6900.52 3 components
Mixing Family: c("NO", "NO", "NO")
Fitting method: EM algorithm
Call: gamlssMX(formula = SR ~ 1, family = NO, K = 3, data = d, plot = FALSE)
Mu Coefficients for model: 1
(Intercept)
5.23
Sigma Coefficients for model: 1
(Intercept)
-0.4888
Mu Coefficients for model: 2
(Intercept)
6.612
Sigma Coefficients for model: 2
(Intercept)
-0.1508
Mu Coefficients for model: 3
(Intercept)
7.271
Sigma Coefficients for model: 3
(Intercept)
-0.263
Estimated probabilities: 0.2537915 0.2147315 0.531477
Degrees of Freedom for the fit: 8 Residual Deg. of Freedom 2257
Global Deviance: 6862.04
AIC: 6878.04
SBC: 6923.84 4 components
Mixing Family: c("NO", "NO", "NO", "NO")
Fitting method: EM algorithm
Call: gamlssMX(formula = SR ~ 1, family = NO, K = 4, data = d, plot = FALSE)
Mu Coefficients for model: 1
(Intercept)
7.171
Sigma Coefficients for model: 1
(Intercept)
-0.2973
Mu Coefficients for model: 2
(Intercept)
5.092
Sigma Coefficients for model: 2
(Intercept)
-0.5485
Mu Coefficients for model: 3
(Intercept)
5.806
Sigma Coefficients for model: 3
(Intercept)
-0.454
Mu Coefficients for model: 4
(Intercept)
7.316
Sigma Coefficients for model: 4
(Intercept)
-0.2168
Estimated probabilities: 0.4443139 0.1849648 0.1507718 0.2199495
Degrees of Freedom for the fit: 11 Residual Deg. of Freedom 2254
Global Deviance: 6861.1
AIC: 6883.1
SBC: 6946.08 5 components
Mixing Family: c("NO", "NO", "NO", "NO", "NO")
Fitting method: EM algorithm
Call: gamlssMX(formula = SR ~ 1, family = NO, K = 5, data = d, plot = FALSE)
Mu Coefficients for model: 1
(Intercept)
6.375
Sigma Coefficients for model: 1
(Intercept)
-0.3485
Mu Coefficients for model: 2
(Intercept)
7.228
Sigma Coefficients for model: 2
(Intercept)
-1.001
Mu Coefficients for model: 3
(Intercept)
7.969
Sigma Coefficients for model: 3
(Intercept)
-0.5569
Mu Coefficients for model: 4
(Intercept)
6.248
Sigma Coefficients for model: 4
(Intercept)
-0.4579
Mu Coefficients for model: 5
(Intercept)
5.041
Sigma Coefficients for model: 5
(Intercept)
-0.6049
Estimated probabilities: 0.1788327 0.2320332 0.1993053 0.2001806 0.1896483
Degrees of Freedom for the fit: 14 Residual Deg. of Freedom 2251
Global Deviance: 6844.22
AIC: 6872.22
SBC: 6952.37 Comparing AIC
df AIC
mix.SR.NO.2 5 6871.892
mix.SR.NO.5 14 6872.218
mix.SR.NO.3 8 6878.040
mix.SR.NO.4 11 6883.103
mix.SR.NO.1 2 6997.163Showing the distributions
Mixture of Gaussians for SR.
IR
Between 1 and 5 Gaussian distributions:
1 component
Mixing Family: "NO"
Fitting method: EM algorithm
Call: gamlssMX(formula = IR ~ 1, family = NO, K = 1, data = d, plot = FALSE)
Mu Coefficients for model: 1
(Intercept)
39.03
Sigma Coefficients for model: 1
(Intercept)
1.598
Estimated probabilities: 1
Degrees of Freedom for the fit: 2 Residual Deg. of Freedom 2263
Global Deviance: 13667
AIC: 13671
SBC: 13682.5 2 components
Mixing Family: c("NO", "NO")
Fitting method: EM algorithm
Call: gamlssMX(formula = IR ~ 1, family = NO, K = 2, data = d, plot = FALSE)
Mu Coefficients for model: 1
(Intercept)
39.76
Sigma Coefficients for model: 1
(Intercept)
1.766
Mu Coefficients for model: 2
(Intercept)
38.36
Sigma Coefficients for model: 2
(Intercept)
1.336
Estimated probabilities: 0.4795038 0.5204962
Degrees of Freedom for the fit: 5 Residual Deg. of Freedom 2260
Global Deviance: 13637.8
AIC: 13647.8
SBC: 13676.4 3 components
Mixing Family: c("NO", "NO", "NO")
Fitting method: EM algorithm
Call: gamlssMX(formula = IR ~ 1, family = NO, K = 3, data = d, plot = FALSE)
Mu Coefficients for model: 1
(Intercept)
35.42
Sigma Coefficients for model: 1
(Intercept)
1.339
Mu Coefficients for model: 2
(Intercept)
39.78
Sigma Coefficients for model: 2
(Intercept)
0.89
Mu Coefficients for model: 3
(Intercept)
42.22
Sigma Coefficients for model: 3
(Intercept)
1.636
Estimated probabilities: 0.3631272 0.2946715 0.3422013
Degrees of Freedom for the fit: 8 Residual Deg. of Freedom 2257
Global Deviance: 13618.4
AIC: 13634.4
SBC: 13680.2 4 components
Mixing Family: c("NO", "NO", "NO", "NO")
Fitting method: EM algorithm
Call: gamlssMX(formula = IR ~ 1, family = NO, K = 4, data = d, plot = FALSE)
Mu Coefficients for model: 1
(Intercept)
42.31
Sigma Coefficients for model: 1
(Intercept)
1.699
Mu Coefficients for model: 2
(Intercept)
39.43
Sigma Coefficients for model: 2
(Intercept)
0.6302
Mu Coefficients for model: 3
(Intercept)
40.07
Sigma Coefficients for model: 3
(Intercept)
1.316
Mu Coefficients for model: 4
(Intercept)
34.48
Sigma Coefficients for model: 4
(Intercept)
1.25
Estimated probabilities: 0.2568272 0.1726828 0.3017301 0.2687599
Degrees of Freedom for the fit: 11 Residual Deg. of Freedom 2254
Global Deviance: 13613.6
AIC: 13635.6
SBC: 13698.6 5 components
Mixing Family: c("NO", "NO", "NO", "NO", "NO")
Fitting method: EM algorithm
Call: gamlssMX(formula = IR ~ 1, family = NO, K = 5, data = d, plot = FALSE)
Mu Coefficients for model: 1
(Intercept)
42.85
Sigma Coefficients for model: 1
(Intercept)
1.694
Mu Coefficients for model: 2
(Intercept)
37.38
Sigma Coefficients for model: 2
(Intercept)
1.457
Mu Coefficients for model: 3
(Intercept)
39.58
Sigma Coefficients for model: 3
(Intercept)
0.4193
Mu Coefficients for model: 4
(Intercept)
40.57
Sigma Coefficients for model: 4
(Intercept)
1.308
Mu Coefficients for model: 5
(Intercept)
35.24
Sigma Coefficients for model: 5
(Intercept)
1.337
Estimated probabilities: 0.2128878 0.2184185 0.1195341 0.2222382 0.2269214
Degrees of Freedom for the fit: 14 Residual Deg. of Freedom 2251
Global Deviance: 13612.4
AIC: 13640.4
SBC: 13720.5 Comparing AIC
df AIC
mix.IR.NO.3 8 13634.37
mix.IR.NO.4 11 13635.64
mix.IR.NO.5 14 13640.36
mix.IR.NO.2 5 13647.78
mix.IR.NO.1 2 13671.04Showing the distributions
Mixture of Gaussians for IR.
Tests of unimodality
We used three ways to estimate how unimodal a distribution is, as they tend to disagree and the problem of unimodality testing is far from settled (see Freeman & Dale, 2013):
- the Silverman test tests the null hypothesis that an underlying density has at most k modes. The null hypothesis is that the underlying density has at most k modes (H0: number of modes <= k). The result is the p-value (bootstrapped) of rejecting a unimodal distribution. It is described in Silverman (1981) and Hall & York (2001); our implementation is based on the code available at https://www.mathematik.uni-marburg.de/~stochastik/R_packages/;
- the dip test which computes Hartigans’ dip statistic D_n_, and its p-value for the test for unimodality, by interpolating tabulated quantiles of sqrt(n) • D_n_. For X_i_ ∼ F, i.i.d., the null hypothesis is that F is a unimodal distribution. The result is the D metric and the p-value (interpolated) of rejecting a unimodal distribution. See Hartigan (1985) and Hartigan & Hartigan (1985); it is implemented in package
diptest; - the bimodality coefficient (BC) which is based on an empirical relationship between bimodality and the third and fourth statistical moments of a distribution (skewness and kurtosis). It is proportional to the division of squared skewness with uncorrected kurtosis, BC ≈ (s2 + 1)/k, with the underlying logic that a bimodal distribution will have very low kurtosis, an asymmetric character, or both; all of these conditions increase BC. The values range from 0 and 1, with those exceeding .555 (the value representing a uniform distribution) suggesting bi-modality. The result is the DC estimate (which must exceed 0.555 to reject a unimodal distribution). We implemented it following Freeman & Dale (2013) as BC = (s2 + 1)/(k + 3 • ((n-1)2 / ((n-2) • (n-3)))).
For each such test, we performed four randomisation procedures to obtain an estimate of the “specialness” of the observed unimodality estimate; for each new permuted dataset, we recompute everything before estimating the unimodlaity of the permuted distribution:
- Permutation model 1 (PM1): randomly permute the SR values freely among speakers, texts and languages;
- Permutation model 2 (PM2): randomly permute the ID values among languages;
- Permutation model 3 (PM3): randomly permute the Speaker average SR values among speakers (irrespective of languages);
- Permutation model 4 (PM4): randomly permute the Language average SR values among languages.
Visual comparison
The observed estimate (vertical blue solid line), the permuted distribution (gray histogram), and the “unimodality region” (shaded green rectangle) are shown below (for PM3, we also show the original estimate using the Speaker average SR as a vertical solid red line).
PM1
Permutation of the texts’ SRs (PM1).
PM2
Permutation of the languages’ ID (PM2).
PM3
Permutation of the speakers’ average SRs (PM3).
PM4
Permutation of the languages’ average SRs with speaker adjustement (PM4).
Summary of unimodality tests
| Scenario | Measure | Test | Observed estimate (p-value) | % more unimodal permutations |
|---|---|---|---|---|
| PM1 | SR | Silverman | - (0.019) | 74.2% |
| PM1 | SR | Dip | 0.006 (0.924) * | 100% |
| PM1 | SR | BC | 0.193 () * | 100% |
| PM1 | IR | Silverman | - (0.17) * | 86.8% |
| PM1 | IR | Dip | 0.005 (0.994) * | 50% |
| PM1 | IR | BC | 0.165 () * | 100% |
| PM2 | SR | Silverman | - (0.019) | 75.2% |
| PM2 | SR | Dip | 0.006 (0.924) * | 100% |
| PM2 | SR | BC | 0.193 () * | 100% |
| PM2 | IR | Silverman | - (0.17) * | 22.6% |
| PM2 | IR | Dip | 0.005 (0.994) * | 14.8% |
| PM2 | IR | BC | 0.165 () * | 97.7% |
| PM3 | SR | Silverman | - (0.019) | 100% |
| PM3 | SR | Dip | 0.006 (0.924) * | 0% |
| PM3 | SR | BC | 0.193 () * | 100% |
| PM3 | IR | Silverman | - (0.17) * | 86.2% |
| PM3 | IR | Dip | 0.005 (0.994) * | 6% |
| PM3 | IR | BC | 0.165 () * | 100% |
| PM4 | SR | Silverman | - (0.019) | 5.2% |
| PM4 | SR | Dip | 0.006 (0.924) * | 3.1% |
| PM4 | SR | BC | 0.193 () * | 75.5% |
| PM4 | IR | Silverman | - (0.17) * | 11.1% |
| PM4 | IR | Dip | 0.005 (0.994) * | 5.9% |
| PM4 | IR | BC | 0.165 () * | 97.3% |
Pair-wise distances between languages
We compute various distances between languages (as implemented by function distance() in package philentropy) in what concerns the distribution of NS, SR and ID.
Comparing the distribution of pairwise distances between languages.
| m1 | m2 | d | mean1 | median1 | sd1 | mean2 | median2 | sd2 | p |
|---|---|---|---|---|---|---|---|---|---|
| IR | NS | Hellinger | 0.89 | 0.84 | 0.32 | 1.20 | 1.19 | 0.39 | 0.00 |
| IR | NS | Jensen-Shannon | 0.18 | 0.14 | 0.12 | 0.29 | 0.26 | 0.16 | 0.00 |
| IR | NS | Kolmogorov–Smirnov | 0.42 | 0.37 | 0.20 | 0.57 | 0.57 | 0.23 | 0.00 |
| IR | NS | Kullback-Leibler | 7.44 | 4.77 | 7.88 | 15.43 | 13.12 | 13.30 | 0.00 |
| IR | NS | Squared-Chi | 0.57 | 0.46 | 0.36 | 0.88 | 0.80 | 0.47 | 0.00 |
| IR | SR | Hellinger | 0.89 | 0.84 | 0.32 | 1.13 | 1.09 | 0.49 | 0.00 |
| IR | SR | Jensen-Shannon | 0.18 | 0.14 | 0.12 | 0.28 | 0.24 | 0.20 | 0.00 |
| IR | SR | Kolmogorov–Smirnov | 0.42 | 0.37 | 0.20 | 0.57 | 0.56 | 0.27 | 0.00 |
| IR | SR | Kullback-Leibler | 7.44 | 4.77 | 7.88 | 15.90 | 10.08 | 16.29 | 0.00 |
| IR | SR | Squared-Chi | 0.57 | 0.46 | 0.36 | 0.88 | 0.78 | 0.58 | 0.00 |
| NS | SR | Hellinger | 1.20 | 1.19 | 0.39 | 1.13 | 1.09 | 0.49 | 0.09 |
| NS | SR | Jensen-Shannon | 0.29 | 0.26 | 0.16 | 0.28 | 0.24 | 0.20 | 0.68 |
| NS | SR | Kolmogorov–Smirnov | 0.57 | 0.57 | 0.23 | 0.57 | 0.56 | 0.27 | 0.99 |
| NS | SR | Kullback-Leibler | 15.43 | 13.12 | 13.30 | 15.90 | 10.08 | 16.29 | 0.73 |
| NS | SR | Squared-Chi | 0.88 | 0.80 | 0.47 | 0.88 | 0.78 | 0.58 | 0.94 |
References
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Appendix I: R session info
This document was compiled on:
R version 3.4.4 (2018-03-15)
**Platform:** x86_64-pc-linux-gnu (64-bit)
locale: LC_CTYPE=en_US.UTF-8, LC_NUMERIC=C, LC_TIME=en_US.UTF-8, LC_COLLATE=en_US.UTF-8, LC_MONETARY=en_US.UTF-8, LC_MESSAGES=en_US.UTF-8, LC_PAPER=en_US.UTF-8, LC_NAME=C, LC_ADDRESS=C, LC_TELEPHONE=C, LC_MEASUREMENT=en_US.UTF-8 and LC_IDENTIFICATION=C
attached base packages: grid, parallel, splines, stats, graphics, grDevices, datasets, utils, methods and base
other attached packages: ggpubr(v.0.2), magrittr(v.1.5), ggrepel(v.0.8.0), RColorBrewer(v.1.1-2), dplyr(v.0.8.0.1), ggridges(v.0.5.1), broman(v.0.68-2), philentropy(v.0.3.0), diptest(v.0.75-7), pander(v.0.6.3), moments(v.0.14), sjPlot(v.2.6.2), sjstats(v.0.17.3), gamlss.mx(v.4.3-5), nnet(v.7.3-12), gamlss(v.5.1-2), nlme(v.3.1-137), gamlss.dist(v.5.1-1), MASS(v.7.3-51.1), gamlss.data(v.5.1-0), lmerTest(v.3.1-0), lme4(v.1.1-20), Matrix(v.1.2-15), plyr(v.1.8.4), reshape2(v.1.4.3), ggplot2(v.3.1.0) and RhpcBLASctl(v.0.18-205)
loaded via a namespace (and not attached): numDeriv(v.2016.8-1), tools(v.3.4.4), TMB(v.1.7.15), backports(v.1.1.3), R6(v.2.4.0), sjlabelled(v.1.0.16), lazyeval(v.0.2.1), colorspace(v.1.4-0), withr(v.2.1.2), tidyselect(v.0.2.5), mnormt(v.1.5-5), emmeans(v.1.3.2), compiler(v.3.4.4), sandwich(v.2.5-0), labeling(v.0.3), scales(v.1.0.0), mvtnorm(v.1.0-8), psych(v.1.8.12), stringr(v.1.4.0), digest(v.0.6.18), foreign(v.0.8-71), minqa(v.1.2.4), rmarkdown(v.1.11), stringdist(v.0.9.5.1), pkgconfig(v.2.0.2), htmltools(v.0.3.6), highr(v.0.7), pwr(v.1.2-2), rlang(v.0.3.1), generics(v.0.0.2), zoo(v.1.8-4), modeltools(v.0.2-22), bayesplot(v.1.6.0), Rcpp(v.1.0.0), munsell(v.0.5.0), prediction(v.0.3.6.2), stringi(v.1.3.1), multcomp(v.1.4-8), yaml(v.2.2.0), snakecase(v.0.9.2), sjmisc(v.2.7.7), forcats(v.0.4.0), crayon(v.1.3.4), lattice(v.0.20-38), ggeffects(v.0.8.0), haven(v.2.1.0), hms(v.0.4.2), knitr(v.1.21), pillar(v.1.3.1), estimability(v.1.3), codetools(v.0.2-16), stats4(v.3.4.4), glue(v.1.3.0), evaluate(v.0.13), data.table(v.1.12.0), modelr(v.0.1.4), nloptr(v.1.2.1), gtable(v.0.2.0), purrr(v.0.3.0), tidyr(v.0.8.2), assertthat(v.0.2.0), xfun(v.0.5), coin(v.1.2-2), xtable(v.1.8-3), broom(v.0.5.1), coda(v.0.19-2), survival(v.2.43-3), tibble(v.2.0.1), glmmTMB(v.0.2.3) and TH.data(v.1.0-10)
Appendix II: Figures for the main paper
Here we generate the figures used in the main paper (saved to the ./figures folder as 600 DPI TIFF files Figure-*.tiff).